08 / CHAPTER 08
Ordinary differential equations
Solve first- and second-order evolution laws.
ANCHOR EQUATION
First-order ODE
In symbols: dy/dt = f(t,y)
An evolution law needs an initial condition to select a trajectory.
An ODE predicts evolution after a state is specified
A differential equation gives a rate law; an initial condition selects which trajectory among many possible solutions is physically realized. The order of the equation counts how many independent conditions are needed.
Integrate a first-order decay
For dy/dt = −ky with k > 0, separate variables dy/y = −kdt. Integrating gives ln|y| = −kt + C, so y = y₀e^{−kt} after applying y(0) = y₀. The exponential appears because the rate is proportional to the amount remaining.
A tank contaminant with k = 0.20 min⁻¹ and y₀ = 50 mg/L falls to y(5) = 50e⁻¹ = 18.4 mg/L. The half-life is ln2/k = 3.47 min.
A rate law is not a curve fit without units
k must have units inverse time for the exponent to be dimensionless. If a source term or nonlinear feedback exists, the exponential limit fails.
Work it: Solve dx/dt = 6 − 0.5x with x(0) = 0. Find equilibrium, time constant, and x at t = 4.0 s by shifting to x − x_eq.